DIFFERENTIAL GEOMETRY: A First Course in Curves and Surfaces

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92 Chapter 3. Surfaces: Further Topics*4. Using Proposition 1.2, prove that horizontal lines have constant geodesic curvature κ g =1inH; prove that this is also true for circles tangent to the u-axis. (These curves are classicallycalled horocycles.)5. Prove that every curve in H of constant geodesic curvature κ g =1is of the form given inExercise 4. (Hints: Assume we start with an arclength parametrization (u(s),v(s)), and useProposition 1.2 to show that we have 1 = u′v + θ′ and u ′2 + v ′2 = v 2 . Obtain the differentialequation(v d2 v( dv) ) 2 3/2 ( dv) 2du 2 = 1+− − 1,duduand solve this by substituting z = dv/du and getting a separable differential equation fordz/dv.)6. Let T a,b,c,d (z) = az + b , a, b, c, d ∈ R, with ad − bc =1.cz + da. Suppose a ′ ,b ′ ,c ′ ,d ′ ∈ R and a ′ d ′ − b ′ c ′ =1. Check thatT a ′ ,b ′ ,c ′ ,d ′◦ T a,b,c,d = T a ′ a+b ′ c,a ′ b+b ′ d,c ′ a+d ′ c,c ′ b+d ′ d(a ′ a + b ′ c)(c ′ b + d ′ d) − (a ′ b + b ′ d)(c ′ a + d ′ c)=1.Show, moreover, that T d,−b,−c,a = T −1a,b,c,d . (Note that T a,b,c,d = T −a,−b,−c,−d . The readerwho’s taken group theory will recognize that we’re defining an isomorphism between thegroup of linear fractional transformations and the group SL(2, R)/{±I} of 2 × 2 matriceswith determinant 1, identifying a matrix and its additive inverse.)b. Let T = T a,b,c,d . Prove that if z = u + iv and v>0, then T (z) =x + iy with y>0.Deduce that T maps H to itself bijectively.7. Show that reflection across the geodesic u =0isgiven by r(z) =−z. Use this to determinethe form of the reflection across a general geodesic.8. The geodesic circle of radius R centered at P is the set of points Q so that d(P, Q) =R. Provethat geodesic circles in H are Euclidean circles. One way to proceed is as follows: The geodesiccircle centered at P =(0, 1) with radius R =lna must pass through (0,a) and (0, 1/a), andhence ought to be a Euclidean circle centered at (0, 1 2(a +1/a)). Check that all the points onthis circle are in fact a hyperbolic distance R away from P . (Hint: It is probably easiest towork with the cartesian equation of the circle.)Remark. This result can be established rather painlessly by using the Poincaré disk modelgiven in Exercise 15.and*9. What is the geodesic curvature of a geodesic circle of radius R in H? (See Exercises 4 and 8.)10. Recall (see, for example, p. 298 and pp. 350–1 of Shifrin’s Abstract Algebra: A GeometricApproach) that the cross ratio of four numbers A, B, P, Q ∈ C ∪ {∞} is defined to be[A : B : P : Q] = Q − A / Q − BP − A P − B .
§2. An Introduction to Hyperbolic Geometry 93a. Show that A, B, P , and Q lie on a line or circle if and only if their cross ratio is a realnumber.b. Prove that if S is a linear fractional transformation with S(A) = 0, S(B) = ∞, andS(P )=1, then S(Q) =[A : B : P : Q]. Use this to deduce that for any linear fractionaltransformation T ,wehave[T (A) :T (B) :T (P ):T (Q)] = [A : B : P : Q].c. Prove that linear fractional transformations map lines and circles to either lines or circles.(For which such transformations do lines necessarily map to lines?)d. Show that if A, B, P , and Q lie on a line or circle, then[A : B : P : Q] = AQ / BQAP BP .Conclude that d(P, Q) =| ln[A : B : P : Q]|, where A, B, P , and Q are as illustrated inFigure 2.2.e. Check that if T is a linear fractional transformation carrying A to 0, B to ∞, P to P ′ ,and Q to Q ′ , then d(P, Q) =d(P ′ ,Q ′ ).11. a. Let O be any point not lying on a circle C and let P and Q be points on the circle C sothat O, P , and Q are collinear. Let T be the point on C so that OT is tangent to C. Provethat (OP)(OQ) =(OT) 2 .b. Define inversion in the circle of radius R centered at O by sending a point P to the pointP ′ on the ray OP with (OP)(OP ′ )=R 2 . Show that an inversion in a circle centered atthe origin maps a circle C centered on the u-axis and not passing through O to anothercircle C ′ centered on the u-axis. (Hint: For any P ∈ C, let Q be the other point on Ccollinear with O and P , and let Q ′ be the image of Q under inversion. Use the result ofpart a to show that OP/OQ ′ is constant. If C is the center of C, let C ′ be the point onthe u-axis so that C ′ Q ′ ‖CP. Show that Q ′ traces out a circle C ′ centered at C ′ .)c. Show that inversion in the circle of radius R centered at O maps vertical lines to circlescentered on the u-axis and passing through O and vice-versa.12. a. Prove that every (orientation-preserving) isometry of H can be written as the compositionof linear fractional transformations of the formT 1 (z) =z + b for some b ∈ R and T 2 (z) =− 1 z .(Hint:[It’s]probably[ ]easiest to work[with]matrices. Show that you have matrices of the1 b 0 −11 0form , , and therefore , and that any matrix of determinant 1 can be0 1 1 0b 1obtained by row and column operations from these.)b. Prove that T 2 maps circles centered on the u-axis and vertical lines to circles centered onthe u-axis and vertical lines (not necessarily respectively). Either do this algebraically oruse Exercise 11.c. Use the results of parts a and b to prove that isometries of H map geodesics to geodesics.13. We say a linear fractional transformation T = T a,b,c,d is elliptic if it has one fixed point, parabolicif it has one fixed point at infinity, and hyperbolic if it has two fixed points at infinity.a. Show that T is elliptic if |a + d| < 2, parabolic if |a + d| =2,and hyperbolic if |a + d| > 2.
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DIFFERENTIALGEOMETRY:A First Course
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4 Chapter 1. Curves2πb2πaFigure 1
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6 Chapter 1. CurvesAlternatively, s
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8 Chapter 1. CurvesAn important obs
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10 Chapter 1. Curvesof the road, st
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12 Chapter 1. CurvesSummarizing,κ(
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14 Chapter 1. Curvescurvature κ. I
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16 Chapter 1. CurvesFigure 2.2Conve
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18 Chapter 1. Curvesalso Exercise 2
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20 Chapter 1. Curvesto the curve β
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22 Chapter 1. Curvesthe osculating
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24 Chapter 1. CurvesWe turn now to
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26 Chapter 1. Curvesintersect C eit
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28 Chapter 1. Curvesh(s,t)α(t)α(s
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30 Chapter 1. Curvesy( x(s) ,y(s))
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32 Chapter 1. Curvesɛ2ɛ 1ɛ 3CFig
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CHAPTER 2Surfaces: Local Theory1. P
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36 Chapter 2. Surfaces: Local Theor
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Page 44 and 45: 42 Chapter 2. Surfaces: Local Theor
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DIFFERENTIAL GEOMETRY: A First Course in Curves and Surfaces

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